Fundamentals of Probability and Statistics for Engineers

176 Fundamentals of Probability and Statistics for EngineersSubstituting Equations (6.34), (6.37), and (6.40) into Equation (6.41) andletting t ! 0 we obtainwhich yieldsdp 1 …0; t†dtˆ p 1 …0; t†‡e t ; p 1 …0; 0† ˆ0; …6:42†p 1 …0; t† ˆte t :…6:43†Continuing in this way we find, for the general term,p k …0; t† ˆ…t†k e t; k ˆ 0; 1; 2; ...: …6:44†k!Equation (6.44) gives the pmf of X(0,t), the number of arrivals duringtime interval [0,t) subject to the assumptions stated above. It is called thePoisson distribution, with parameters and t. However, since and t appear inEquation (6.44) as a product, t, it can be replaced by a single parameter , ˆ t, and so we can also writep k …0; t† ˆk e ;k!k ˆ 0; 1; 2; ...: …6:45†The mean of X(0, t) is given byEfX…0; t†g ˆ X1kˆ0ˆ tekp k …0; t† ˆetX1kˆ1tX1kˆ0k…t† kk!…t† k 1…k 1†! ˆ te t e t ˆ t:…6:46†Similarly, we can show that 2 X…0;t† ˆ t:…6:47†It is seen from Equation (6.46) that parameter is equal to the averagenumber of arrivals per unit interval of time; the name ‘mean rate of arrival’ for, as mentioned earlier, is thus justified. In determining the value of thisparameter in a given problem, it can be estimated from observations by m/n,TLFeBOOK